Abstract

A modelocked fiber laser, operating in the soliton regime without any explicit intracavity
polarizers, is observed to spontaneously lock its output polarization for certain values of the
intracavity birefringence. For other settings of the intracavity birefringence the output
polarization undergoes pulse-to-pulse evolution. The dependence of the output polarization evolution
on intracavity birefringence outside of the locking regions can be understood with a simple model.
The locking behavior exhibits several surprising aspects and is not completely understood.

] However modelocked fiber
lasers can also display interesting and complex dynamics, particularly when they are operated with
net anomalous dispersion in the cavity so that the pulse exhibits soliton-like behavior. In this
regime, the group-velocity-dispersion (GVD) is canceled by the self-phase-modulation (SPM), yielding
a stable, circulating pulse in the cavity.[3

] that is modelocked
using a semiconductor saturable absorber and operating in the soliton regime. This laser is
different from most others in that it does not contain any explicit polarizing elements or
polarization maintaining fiber. The polarization is then free to evolve under the combined
influences of any weak, strain-induced fiber birefringence and nonlinear polarization rotation. When
the laser is modelocked, we observe that for certain values of net cavity birefringence that the
output polarization stops evolving, i.e., the output polarization is “locked”. This
behavior is attributed to the interplay between linear birefringence and nonlinear polarization
rotation, although the detailed stability mechanisms remain unclear.

Fig. 2 RF spectra without (a) and with (b) the external linear polarizer. Δ denotes the
polarization evolution frequency (PEF). (c) shows how the pulse-to-pulse evolution of the
polarization ellipse is mapped into amplitude modulation , t is the cavity round-trip time.

2. Laser Cavity and Measurement

The laser is a simple linear cavity consisting of three pieces of fiber fusion-spliced together
(see Fig. 1). The central piece is Er/Yb co-doped gain
fiber, approximately 15 cm in length. A 30 cm piece of standard single mode fiber (SMF) is spliced
to one end of the gain fiber. The other end of this piece of SMF is connectorized and a dielectric
output coupler is deposited directly on the face of the ferrule (~1% transmission at 1550 nm). In
addition, the pump light from a 980 nm telecom grade laser diode is coupled into the cavity through
this mirror, which has high transmission at 980 nm. The other side of the gain fiber is spliced to a
~155 cm piece of SMF that is butt-coupled to a Saturable Bragg Reflector (SBR). [8–10

] The SBR consists of an AlAs/GaAs Bragg high reflector
with a pair of InGaAs/InP quantum wells grown in the top layer. The quantum wells provide saturable
absorption that modelocks the laser. The 155 cm piece of SMF is wrapped around 2 paddles (5.6 cm
radius, 3 wraps each) of a standard fiber polarization controller. [11

To measure the evolution of the polarization of the output pulses, we monitor the RF spectrum
after passing the pulses through a linear polarizer. RF spectra with and without the linear
polarizer are shown in Fig. 2. Without the polarizer the RF
spectrum consists simply of the comb of harmonics corresponding to the repetition rate (Fig. 2a). With the linear polarizer in place, strong sidebands
appear in the spectrum (Fig. 2b). These sidebands can
readily be understood as arising from the evolution of the pulse polarization during each round
trip. The cavity birefringence transforms the polarization state of the pulse during a round trip.
The linear polarizer selects a given axis of the polarization ellipse, mapping the polarization
evolution into amplitude modulation (see Fig. 2c). The
amplitude modulation results in the observed sidebands in the RF spectrum, with spacing denoted by
Δ in Fig. 2. The spacing of the sidebands from
cavity harmonics is a measure of the rate at which the polarization evolves, hence we denote it as
the polarization evolution frequency (PEF). The PEF is inversely proportional to the birefringence
beat length Lb, i.e., Lb = Lc/(Δτ), where Lc is the length of a cavity round-trip and τ is cavity round trip time. Note that
the measured PEF is the real REF modulo the half of the cavity repetition rate since we only sample
the polarization once per round-trip (at the output coupler). If the polarization evolution
undergoes greater than one complete cycle per cavity round-trip it is aliased to a lower frequency
in our measurement. If aliasing does occur, the relationship between PEF and beat length no longer
holds.

Fig. 3 Polarization evolution frequency as function of the angles of the two polarization controller
paddles for CW operation of the laser.

3. Linear Operation

To obtain an understanding of the polarization evolution in the laser cavity without the
complication of nonlinear polarization rotation due to Kerr nonlinearities, we first replace the SBR
with a dielectric high reflector. In this case the laser runs CW. Although the sidebands are not
nearly as well defined as for modelocked operation, it is still possible to determine the PEF. In
Fig. 3 we show a map of the PEF as a function of the angles
of the two polarization controller paddles (Θ1 & Θ2 in
Fig. 1). The polarization controller paddles essentially act
as waveplates, where the birefringence is due to bending induced strain. [11

] We model the cavity as a series of three bulk waveplates, with adjustable
retardance and angle. For each round trip through the cavity each waveplate is transited twice, in
the correct sequence. The cavity is unfolded, so that the flip in handedness upon reflection is
canceled by transiting the waveplate in the opposite direction. Two of the three waveplates model
the two polarization controller paddles, the third represents the residual birefringence elsewhere
in the cavity.

To obtain the PEF, a transfer matrix for a single round-trip through the cavity is calculated for
a given setting of the waveplates. An arbitrary initial polarization state is chosen. The transfer
matrix is then applied 512 times, and the intensity along one linear polarization state is recorded
after each application. These 512 intensities are taken as a time series, the average values
subtracted out (to remove the DC term), and a Fourier transform performed to obtain the RF spectrum.
The PEF is then defined as the strongest channel in the spectrum. The PEF does not depend on the
arbitrary initial polarization state, but the amplitude of the resulting modulation does.

Fig. 4 Simulated PEF(Θ1,Θ2). ϕ12 is the
retardance of the two polarization controllers, ϕR is the residual
retardance.

As a starting point we first simulate the case where the two paddles of the polarization
controller are quarter-wave-plates and there is no residual birefringence in the cavity. An estimate
of the retardance based on Ref. 11

shows that a quarter wave
of retardance should be a good estimate for the geometry of our controller. However these parameters
yield a PEF(Θ1,Θ2) map that consists of diagonal stripes
(Fig. 4a), i.e. not reproducing the experiment. Since there
is no absolute angle reference, it can easily be understood that only the relative angle between the
two paddles is relevant.

The need for an absolute angle reference suggests the inclusion of a weak waveplate with fixed
angle to model the residual birefringence (a retardance of ϕR) elsewhere in the
cavity. However this alone does not provide qualitative reproduction of the experiment (Fig. 4b). Only upon adjustment of the magnitude of the retardance
for the polarization controller paddles (ϕ12, they are always taken to be
identical) do we obtain satisfactory agreement with the experiment (Fig. 4d). To demonstrate that both ingredients are necessary to reproduce the experiment,
the result for only adjusting the value of ϕ12 is shown in Fig. 4c. We find that ϕ12 = 1.2 radians and
ϕR = 0.5 radians provides good agreement.

5. Modelocked Operation

With the SBR in the cavity, modelocked (ML) operation is obtained with an output pulse width of
approximately 400 fs. The laser shows all of the hallmarks of soliton operation: the optical
spectrum is well fit by a sech2, it displays spectral sidebands due to periodic
perturbation of the soliton by the cavity [14

The PEF(Θ1,Θ2) for ML operation is shown in Fig. 5a. These results show a dramatic new feature that is not
present for CW operation, namely the regions where the polarization “locks”, i.e. it
does not evolve. These regions are shown as black in Fig. 5.
It is important to note that in these regions the amplitude of the sidebands vanishes (it drops
below the noise floor of the measurement, which is 25–30 dB down). All of these data were
taken with careful control to maintai n the same optical spectrum.

Fig. 6 High resolution scan of PEF vs. Θ2 for fixed Θ1 =
110°.

Fig. 7 Optical (left) and corresponding RF (right) spectra for 3 different settings of Q1 and
Q2. For the top two panels the polarization is not locked, for the bottom panel it is
locked. In the optical spectra, CSB denotes sidebands that are due to the period perturbation by the
cavity, while PSB denotes sidebands due to the period evolution of the polarization over several
cavity round trips. The peak marked by C in the RF spectra is the fundamental repetition rate, while
those marked by P are due to polarization evolution. The small peaks in the polarization locked case
is due to mode beating in the pump laser, not polarization evolution.

A higher resolution map of the PEF(Θ1,Θ2) in the vicinity
of locking reveals that there are actually several distinct locking regions (Fig. 5b). One of these regions occurs near where the PEF naturally goes to zero
(as estimated from the nearby non-locking behavior, or from the CW operation, it is near the center
of the elliptically shaped region). In this region one might argue that a small
polarization-dependent loss could overcome the small polarization ellipse rotation that occurs due
to the birefringence experienced during each round trip. However there are several other areas that
also display locking, although the “natural” PEF is clearly non-zero.

Several aspects of the locking regions can be seen more clearly in a high resolution, one
dimensional scan. In Fig. 6, the results are shown for
scanning Θ2 while holding Θ1 fixed at 110°. In this
figure we plot PEF = 0 in the locked regions. The three locking regions are clearly evident. It also
clear that the transition between locked and unlocked regions is very abrupt, it occurs for a 1
degree change in angle, which is essentially the resolution of the setup. The PEF for CW and ML
operation track very well, except for the regions where polarization locking occurs.

Examination of the ML data near Θ2 = 40° shows that a locking behavior
occurs here as well. This is manifest as a several degree region for which the PEF stays constant at
exactly half the laser repetition rate, i.e. this corresponds to a period doubling behavior. Period
tripling can be observed near Θ2 = 30°, although over a narrow range
(only 1–2°). It is interesting to note that the period doubling and tripling locking
only occur at the values of Θ2 mentioned, and do not occur at the other values of
Θ2 for which the PEF goes through these frequencies.

The final feature observed in Fig. 6 is the presence of a
small region for which there is no data for ML operation (near 100°, at the edge of the
polarization locking region). In this region the laser does not produce stable modelocking. We
speculate that the intracavity birefringence is such that it generates close to a linear
polarization state in the gain medium, resulting in polarization hole burning which destabilizes the
modelocking. In general, we discount polarization hole burning as a mechanism for the polarization
locking because it should be present in CW operation as well, and we never observe CW polarization
locking. Ultrafast hole burning could play a role in the observed behavior, and may be present as
spectral holes with a homogeneous width of approximately 10 nm have been observed in erbium
amplifiers. [16

In Fig. 7, the optical and RF spectra are shown for 3
settings of the polarization controller paddles. In (a) and (b), the polarization is not locked,
while in (c), it is locked. The sidebands due to the periodic perturbation of the soliton by the
cavity is observable in all three optical spectra (marked by CSB in the figure). In (a) and (b),
where the polarization is not locked, additional sidebands are present. These sidebands correspond
to a periodic perturbation with a period of several cavity round trips, i.e., the time it takes for
the polarization to undergo a complete cycle. We therefore designate these as polarization sidebands
(PSB) in Fig. 7. The position of these sidebands does depend
on the settings of the polarization controller paddles (compare (a) and (b)) as expected. The PSB
spacing decreases with decreasing PEF because a smaller PEF corresponds to a perturbation with a
longer period. Additionally, (c) shows that they disappear when the polarization locks. Note that,
in the RF spectrum (c), the remaining peaks are not due to polarization evolution (they are not
properly spaced), but rather due to mode beating in the 980 nm pump laser.

Feedback into a laser cavity can dramatically effect both modelocking and polarization dynamics.
We have observed these effects for several different configurations of the components external to
the cavity, including launching the laser output into a fiber amplifier with an optical isolator. We
therefore conclude that feedback does not play a role in the observed polarization locking.

6. Output Polarization

We have measured the output polarization state in the locked regions. The measurements were
performed using a Hewlett-Packard 8509B Polarization Analyzer. A fiber polarization controller
between the laser output coupler and polarization analyzer was used to null out the birefringence of
the intervening fiber pigtails. We confirmed the proper setting of the polarization controller by
free space coupling linearly polarized light in place of the laser output.

We observe that, in the locking regions that correspond to a “natural” zero of
the PEF, the laser output is elliptically polarized. In the other regions, where the PEF is
“naturally” non-zero, the output is nearly linearly polarized. (The small residual
ellipticity is within that expected due to imperfect nulling of the pigtail birefringence.) In the
latter regions, the axis of linear polarization systematically rotates from one side of the region
to the other. The output polarization returns to the same state after cycling the pump power.
Additionally the polarization state does not depend on the details of the pulse spectrum (as long as
the polarization remains locked).

7. Discussion

The polarization locking is clearly a nonlinear process as it only occurs when the laser is
modelocked. The locking regions are sensitive to the spectral width, which is probably due to the
concomitant change in the soliton pulse width. All of the data shown above were taken with careful
control to maintain the same spectral width throughout. A simple extension of the simulation
described in section 4 to include Kerr nonlinearities as a lumped element (nonlinear polarization
rotation) does not give locking behavior. This is not surprising as it is well known that
nonlinearity and birefringence must be handled simultaneously. [17

A more complete analysis requires the solution of coupled nonlinear Schrödinger equations
to describe the propagation of the solitons under the combined influences of birefringence and
nonlinearity. Analytical and numerical studies of soliton propagation in birefringent fiber have
concluded that the Kerr nonlinearity can cancel the effect of birefringence, locking the orthogonal
states together. [18–20

] These results are related to the idea of a vector soliton, i.e. a soliton that maintains
not only its intensity profile but also its polarization state, even if it is not propagating along
one of the principal axes of the fiber. [3

8. Conclusion

We have presented experimental results that demonstrate the spontaneous locking of the output
polarization state from an isotropic, modelocked fiber laser operating in the soliton regime. A
simple simulation provides good reproduction of the polarization evolution when the laser is running
CW. The polarization locking only occurs in the modelocked regime and is clearly due to fiber
nonlinearity. A detailed understanding of the locking mechanisms requires a more extensive and
sophisticated model and/or analysis.

Acknowledgments

The authors thank J.E. Cunningham and W.Y Jan for providing the SBR, D.J. DiGiovanni for
providing the Er/Yb fiber, G.T. Harvey for the loan of the polarization analyzer, E.P. Ippen for
helpful discussions and L.E. Nelson for a critical reading of the manuscript.

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